Let $P$ and $Q$ be $3 imes 3$ matrices with $P
eq Q$. If $P^3=Q^3$ and $P^2 Q=Q^2 P$, then determinant of $\left(P^2+Q^2 ight)$ is equal to

Question

Let $P$ and $Q$ be $3 	imes 3$ matrices with $P 
eq Q$. If $P^3=Q^3$ and $P^2 Q=Q^2 P$, then determinant of $\left(P^2+Q^2ight)$ is equal to

Aicharya · Accepted Answer

$P^3=Q^3$ $P^3-P^2 Q=Q^3-Q^2 P$ $P^2(P-Q)=Q^2(Q-P)$ $P^2(P-Q)+Q^2(P-Q)=O$ $\left(P^2+Q^2\right)(P-Q)=O \quad \Rightarrow\left|P^2+Q^2\right|=0$

Let $P$ and $Q$ be $3 \times 3$ matrices with $P \neq Q$. If $P^3=Q^3$ and $P^2 Q=Q^2 P$, then determinant…